A269361 -id:A269361 - cp's OEIS frontend

A269366 a(1) = 1, a(2n) = A269361(1+a(n)), a(2n+1) = A269363(a(n)).

1, 2, 3, 4, 6, 5, 7, 8, 12, 16, 15, 9, 11, 10, 22, 13, 24, 18, 38, 40, 48, 33, 46, 20, 14, 32, 27, 17, 19, 25, 44, 29, 28, 50, 75, 21, 30, 72, 71, 73, 70, 133, 139, 113, 76, 129, 91, 42, 35, 36, 23, 37, 54, 45, 51, 26, 43, 49, 39, 82, 62, 128, 107, 80, 56, 53, 83, 114, 140, 109, 214, 52, 59, 34, 47, 149, 150, 141, 123, 221, 111, 121

Offset: 1

Antti Karttunen, Feb 25 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A269361(1+n), and each right hand child as A269363(n), when the parent node contains n:
                                    |
                 ...................1...................
                2                                       3
      4......../ \........6                   5......../ \........7
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  8       12         16       15          9       11         10       22
13 24   18  38     40  48   33  46      20 14   32  27     17  19   25  44
etc.
An example of (suspected) "entanglement permutation" where the other pair of complementary sequences is generated by a greedy algorithm.
Sequence is not only injective, but also surjective on N (thus a permutation of natural numbers) provided that A269361 is surjective on A091072 and A269363 is surjective on A091067.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Left inverse: A269365 (also right inverse, if this sequence is a permutation of natural numbers).

Cf. A091067, A091072, A269361, A269363, A269367, A266121.

A266121 Lexicographically first injection of natural numbers beginning with a(1)=1 such that 1+(a(n)*a(n+1)) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Original entry on oeis.org

1, 3, 5, 4, 2, 8, 9, 7, 12, 6, 14, 24, 11, 13, 20, 16, 10, 26, 40, 17, 15, 39, 28, 19, 27, 25, 23, 48, 22, 30, 44, 31, 33, 32, 18, 36, 29, 47, 45, 52, 21, 55, 49, 84, 61, 43, 51, 53, 80, 34, 64, 37, 35, 59, 75, 117, 93, 91, 57, 41, 100, 82, 50, 104, 42, 98, 106, 90, 114, 72, 58, 144, 65, 63, 151, 56, 38, 54, 76, 71, 60

Offset: 1

Antti Karttunen, Dec 23 2015

It is conjectured that this sequence is not only injective, but also surjective on N, i.e., that it is a true permutation of natural numbers.
A similar sequence, but with condition that "(a(n)*a(n+1)) must be a member of A003714" yields a sequence: 1, 2, 4, 5, 8, 9, 16, 10, 13, 20, ... (A269361) which certainly is not a bijection, because it contains only terms of A091072.
Also, with above condition and the initial value a(1) = 3 the algorithm generates A269363 which contains only terms of A091067. See also A266191.

			After the initial a(1) = 1, for obtaining the value of a(2) we try the first unused number, which is 2, but (1*2)+1 = 3, which in binary is "11", thus 2 is not qualified at this point of time. So next we try 3, and (1*3)+1 = 4, which in binary is "100", and that satisfies our criterion (no adjacent 1-bits), thus we set a(2) = 3.
For a(3), we test with the least unused numbers 2, 4, 5, etc., yielding products (3*2)+1 = 7 = "111", (3*4)+1 = 13 = "1101" and (3*5)+1 = 16 = "10000" in binary, and only 5 satisfies the criterion, thus we set a(3) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Left inverse: A266122 (also the right inverse if this sequence is a permutation of natural numbers).
Cf. A003714, A003987, A091067, A091072, A269361, A269363, A269365, A269366, A269367.
Cf. also A266191 and A266117 for similar permutations.

Minor typo in the description corrected by Antti Karttunen, Feb 25 2016

A269363 Lexicographically first injection of natural numbers beginning with a(1)=3 such that for all n >= 1, a(n)*a(n+1) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Original entry on oeis.org

3, 6, 7, 12, 11, 15, 22, 24, 14, 19, 27, 38, 28, 23, 46, 48, 43, 30, 39, 35, 59, 44, 31, 75, 62, 87, 51, 83, 56, 47, 88, 54, 76, 55, 96, 86, 60, 71, 67, 70, 78, 112, 79, 107, 102, 91, 120, 139, 118, 140, 119, 142, 131, 134, 155, 240, 156, 135, 152, 108, 95, 92, 103, 179, 184, 115, 147, 224, 94, 175, 123, 150, 111, 158, 214, 163, 203

Offset: 1

Antti Karttunen, Feb 25 2016

The sequence is conjectured to be a permutation of A091067.

The scatter plot is quite interesting (essentially the same as A269367). Compare also to the graph of A269361.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003714, A003987, A091067, A266121, A266191, A269361, A269365, A269366.

Cf. A269367 (the terms ranked with A255070).

fibbinaryQ[n_] := BitAnd[n, 2 n]==0; a[1]=3; a[n_] := a[n] = For[k=1, True, k++, If[Mod[k, 4] != 1, If[fibbinaryQ[a[n-1] k], If[FreeQ[Array[a, n-1], k], Return[k]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2016 *)

A269367 Suspected permutation of natural numbers: a(n) = A255070(A269363(n)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 11, 6, 8, 12, 17, 13, 10, 21, 23, 19, 14, 18, 16, 28, 20, 15, 35, 30, 41, 24, 39, 27, 22, 42, 25, 36, 26, 47, 40, 29, 34, 32, 33, 37, 55, 38, 51, 49, 43, 59, 67, 57, 68, 58, 69, 64, 65, 75, 119, 76, 66, 74, 52, 46, 44, 50, 87, 90, 56, 71, 111, 45, 85, 60, 72, 54, 77, 104, 79, 99, 88, 95, 48, 53, 78, 102, 82, 31

Offset: 1

Antti Karttunen, Feb 27 2016

Lexicographically first injection of natural numbers beginning with a(1)=1 such that for all n >= 1, A091067(a(n))*A091067(a(n+1)) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Antti Karttunen, Table of n, a(n) for n = 1..13217
Index entries for sequences that are permutations of the natural numbers

Left inverse: A269368 (also the right inverse if this is a permutation of natural numbers).

Cf. A003714, A091067, A266121, A269361, A269363, A255070.

a(n) = A255070(A269363(n)).

A306992 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is digitally balanced.

Original entry on oeis.org

1, 2, 5, 7, 6, 25, 9, 15, 10, 17, 12, 13, 4, 3, 14, 11, 18, 31, 19, 29, 8, 23, 26, 22, 40, 63, 33, 69, 36, 67, 34, 21, 99, 27, 30, 82, 28, 81, 35, 61, 16, 47, 50, 53, 48, 73, 37, 20, 41, 56, 45, 49, 43, 52, 51, 42, 54, 46, 55, 44, 57, 38, 65, 127, 66, 135, 70

Offset: 1

Rémy Sigrist, Mar 18 2019

Digitally balanced numbers correspond to A031443.

This sequence has similarities with A269361.

			The first terms, alongside the binary representation of a(n)*a(n+1), are:
  n   a(n)  bin(a(n)*a(n+1))
  --  ----  ----------------
   1     1                10
   2     2              1010
   3     5            100011
   4     7            101010
   5     6          10010110
   6    25          11100001
   7     9          10000111
   8    15          10010110
   9    10          10101010
  10    17          11001100
  11    12          10011100
  12    13            110100
  13     4              1100
  14     3            101010
  15    14          10011010
  16    11          11000110

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A306992

See A306994 for the additive variant.

Cf. A031443, A269361.

PARI
```
See Links section.
```

A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

Original entry on oeis.org

1, 3, 2, 6, 4, 7, 8, 12, 5, 11, 9, 14, 16, 15, 13, 19, 21, 22, 10, 23, 17, 24, 18, 27, 29, 30, 26, 35, 52, 38, 25, 31, 32, 28, 33, 47, 34, 46, 20, 39, 40, 44, 36, 43, 37, 48, 41, 75, 53, 59, 61, 54, 57, 55, 58, 60, 64, 51, 65, 56, 66, 62, 50, 71, 45, 78, 42

Offset: 1

Rémy Sigrist, Jun 28 2017

A144795 gives the numbers without isolated 1's in base-2 representation.
This sequence is conjectured to be a permutation of the natural numbers.
This sequence has similarities with A269361: here we require that the product of two consecutive terms has no isolated 1, there the product of two consecutive terms has only isolated 1's, in base-2 representation.
For any k > 0:
- a(2*k-1) belongs to A091072,
- a(2*k) belongs to A091067.

			The first terms, alongside a(n)*a(n+1) in binary, are:
n       a(n)    a(n)*a(n+1) in binary
--      ----    ---------------------
1       1       11
2       3       110
3       2       1100
4       6       11000
5       4       11100
6       7       111000
7       8       1100000
8       12      111100
9       5       110111
10      11      1100011
11      9       1111110
12      14      11100000
13      16      11110000
14      15      11000011
15      13      11110111
16      19      110001111
17      21      111001110
18      22      11011100
19      10      11100110
20      23      110000111

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A289194

Cf. A091067, A091072, A144795, A269361.

a = {1}; Do[k = 1; While[Nand[! MemberQ[a, k], ! MemberQ[Length /@ DeleteCases[Split[IntegerDigits[k Last[a], 2]], s_ /; First@ s == 0], 1]], k++]; AppendTo[a, k], {n, 2, 67}]; a (* Michael De Vlieger, Jun 29 2017 *)

cp's OEIS Frontend

A269366 a(1) = 1, a(2n) = A269361(1+a(n)), a(2n+1) = A269363(a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

A266121 Lexicographically first injection of natural numbers beginning with a(1)=1 such that 1+(a(n)*a(n+1)) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A269363 Lexicographically first injection of natural numbers beginning with a(1)=3 such that for all n >= 1, a(n)*a(n+1) is a fibbinary number (A003714), i.e., has no adjacent 1's in its base-2 representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A269367 Suspected permutation of natural numbers: a(n) = A255070(A269363(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A306992 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is digitally balanced.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica